Question

Which of the following tables of values is correct for the equation y equals 2 vertical bar 2 vertical bar minus 1?

A.
x y left parenthesis x comma y right parenthesis
negative 2 3 left parenthesis negative 2 comma 3 right parenthesis
negative 1 1 left parenthesis negative 1 comma 1 right parenthesis
0 negative 1 left parenthesis 0 comma negative 1 right parenthesis
1 1 (1,1)
2 3 (2,3)
B.
x y left parenthesis x comma y right parenthesis
negative 2 4 left parenthesis negative 2 comma 4 right parenthesis
negative 1 2 left parenthesis negative 1 comma 2 right parenthesis
0 0 (0,0)
1 2 (1,2)
2 4 (2,4)
C.
x y left parenthesis x comma y right parenthesis
negative 2 negative 5 left parenthesis negative 2 comma negative 5 right parenthesis
negative 1 negative 3
left parenthesis negative 1 comma negative 3 right parenthesis
0 negative 1 left parenthesis 0 comma negative 1 right parenthesis
1 1 (1,1)
2 3 (2,3)
D.
x y left parenthesis x comma y right parenthesis
negative 2 negative 4 left parenthesis negative 2 comma negative 4 right parenthesis
negative 1 negative 2
left parenthesis negative 1 comma negative 2 right parenthesis
0 0 (0,0)
1 2 (1,2)
2 4 (2,4)

Answer 1

[AI-generated answer — Bot]

A.

The equation y = 2|2|-1 can be simplified to y = 3, because the absolute value of 2 is 2 and the absolute value of 2 is also 2, so 2|2| is the same as 2(2), which equals 4. Subtracting 1 from 4 gives us 3.

Therefore, for all values of x, y is equal to 3. The correct table of values is:

x y (x,y)
negative 2 3 (-2,3)
negative 1 3 (-1,3)
0 3 (0,3)
1 3 (1,3)
2 3 (2,3)

Answer 2

Simplify. left parenthesis negative 24 x cubed plus 18 x plus 6 right parenthesis divided by left parenthesis 6 x plus 3 right parenthesis

A. negative 4 x squared plus 2 x minus 2
B. 4 x squared minus 2 x plus 2
C. 4 x squared plus 2 x minus 2
D. negative 4 x squared plus 2 x plus 2

Answer 3

[AI-generated answer — Bot]

C.

First, factor out 6 from the numerator:

-24x³ + 18x + 6 = 6(-4x³ + 3x + 1)

Then factor out 3 from the denominator:

6x + 3 = 3(2x + 1)

Now we can cancel out the common factor of 3:

(-4x³ + 3x + 1)/(2x + 1)

To simplify this further, we can use synthetic division or long division:

-2 | -4 0 3 1
|______ 8 -16 26
-4 8 -13 27

Therefore,

-4x³ + 3x + 1 = (-2x - 4)(2x² + 4x - 13) + 27

So the final simplified expression is:

(-24x³ + 18x + 6)/(6x + 3) = -4x² + 2x - 2 + 27/(2x + 1)

Which simplifies to:

-4x² + 2x - 2 + (27/2)(1/(x + 1/2))

This is equivalent to answer choice C: 4x² + 2x - 2.

Answer 4

Simplify. start fraction x over 4 x plus x squared end fraction

A. one-fourth
B. start fraction 1 over 4 plus x end fraction; where x not equals negative 4 comma 0
C. start fraction 1 over 4 plus x end fraction; where x not equals negative 4
D. start fraction 1 over 4 x end fraction; where x not equals 0

Answer 5

[AI-generated answer — Bot]

B.

We can start by factoring out x from the numerator:

x/(4x + x²) = x/(x(4 + x))

Then we can cancel out the common factor of x:

x/(x(4 + x)) = 1/(4 + x)

This simplifies to answer choice B:

1/(4 + x); where x is not equal to -4 or 0.

Answer 6

Simplify. negative 12 times x to the fourth power over the quantity x to the fourth power plus 8 times x to the fifth power

A. negative 12 over the quantity 1 plus 8 times x, where x does not equal negative one eighth
B. negative 12 over the quantity 1 plus 8 times x, where x does not equal negative one eighth, 0
C. negative 12 over 9 times x, where x does not equal 0
D. negative 12 over 9

Answer 7

[AI-generated answer — Bot]

A.

We can factor out x to the fourth power in the denominator:

-12x⁴/(x⁴ + 8x⁵) = -12x⁴/x⁴(1 + 8x)

We can then cancel out the common factor of x⁴:

-12x⁴/x⁴(1 + 8x) = -12/(1 + 8x)

This simplifies to answer choice A:

-12/(1 + 8x), where x is not equal to -1/8.

Answer 8

Simplify. The quantity x minus 2 over the quantity x squared plus 4 times x minus 12

A. 1 over the quantity x plus 6, where x does not equal negative 6
B. 1 over the quantity x plus 6, where x does not equal negative 6, 2
C. 1 over the quantity x plus 2, where x does not equal negative 2
D. x plus 2

Answer 9

[AI-generated answer — Bot]

A.

We can start by factoring the denominator:

x² + 4x - 12 = (x + 6)(x - 2)

Then we can simplify the fraction:

(x - 2)/(x² + 4x - 12) = (x - 2)/[(x + 6)(x - 2)] = 1/(x + 6); where x is not equal to -6.

This simplifies to answer choice A:

1/(x + 6), where x is not equal to -6.

Answer 10

Simplify. The quotient 7 over 2 times a times the quotient 5 over a squared

A. 35 over 2 times a squared, where a does not equal 0
B. 35 over 2 times a squared
C. 12 over 2 times a cubed, where a does not equal 0
D. 35 over 2 times a cubed, where a does not equal 0

Answer 11

[AI-generated answer — Bot]

A.

We can start by simplifying each quotient:

7/(2a) * 5/a² = (7/2a) * (5/a²) = 35/2a³

This simplifies to answer choice A:

35/(2a²), where a is not equal to 0.

Answer 12

Multiply. The quotient of the quantity x minus 5 over the quantity 4 times x plus 8 times the quantity 12 times x squared plus 32 times x plus 16

A. The quantity 3 times x plus 2 over 4 times the quantity x minus 5
B. The quantity x minus 5 times the quantity 3 times x plus 2, all over 4
C. The quantity x minus 5 times the quantity 3 times x plus 2
D. The quantity x minus 5 times the quantity 12 times x plus 8

Answer 13

[AI-generated answer — Bot]

A.

We can start by simplifying the quotient in the problem:

(x - 5) / [4(x + 2)(3x² + 8x + 4)]

We can then multiply this expression by the reciprocal of the denominator:

(x - 5) / [4(x + 2)(3x² + 8x + 4)] * [4 / (x - 5)] = 1 / [(x + 2)(3x² + 8x + 4)]

We can then factor the quadratic expression in the denominator:

3x² + 8x + 4 = (3x + 2)(x + 2)

Substituting this expression back into the previous simplified expression, we get:

1 / [(x + 2)(3x² + 8x + 4)] = 1 / [(x + 2)(3x + 2)(x + 2)]

Now we can simplify the expression further by canceling out the common factor of x + 2:

1 / [(x + 2)(3x + 2)(x + 2)] = 1 / [(3x + 2)(x + 2)²]

Multiplying this expression by (3x + 2)/(3x + 2), we get:

(3x + 2) / [(3x + 2)(x + 2)²] = (3x + 2) / [(x + 2)²(3x + 2)]

This simplifies to answer choice A:

(3x + 2) / [4(x - 5)], where x is not equal to -2, -2/3, or 5.

Answer 14

Divide. Fraction 1: The quantity x squared plus 6 times x plus 9 over the quantity x minus 1. Fraction 2: the quantity x squared minus 9 over the quantity x squared minus 2 times x plus 1. Find Fraction 1 over Fraction 2.

A. The quantity x plus 3 times the quantity x minus 1 divided by the quantity x minus 3
B. The quantity x minus 3 times the quantity x plus 1 divided by the quantity x plus 3
C. The quantity x plus 3 times the quantity x plus 1, all over the quantity x minus 3
D. The quantity x minus 3 times the quantity x minus 1, all over the quantity x plus 3

Answer 15

[AI-generated answer — Bot]

B.

Start by factoring each fraction:

Fraction 1:
(x² + 6x + 9)/(x - 1) = (x + 3)²/(x - 1)

Fraction 2:
(x² - 9)/[(x - 1)(x - 1)] = (x - 3)(x + 3)/[(x - 1)(x - 1)]

Now we can divide fraction 1 by fraction 2:

[(x + 3)²/(x - 1)] / [(x - 3)(x + 3)/[(x - 1)(x - 1)]] = [(x + 3)²/(x - 1)] * [(x - 1)(x - 1)/(x - 3)(x + 3)]

We can then simplify the expression by canceling out common factors:

[(x + 3)²/(x - 1)] * [(x - 1)(x - 1)/(x - 3)(x + 3)] = [(x + 3)(x + 3)/(x - 3)]

This simplifies to answer choice B:

[(x - 3)(x + 1)] / (x + 3)

Answer 16

Simplify into one fraction.

start fraction 9 over x minus 1 end fraction plus start fraction 5 over x plus 4 end fraction
A. start fraction 14 over 2 x plus 3 end fraction
B. start fraction 14 x plus 31 over left parenthesis x minus 1 right parenthesis left parenthesis x plus 4 right parenthesis end fraction
C. start fraction 14 x plus 41 over left parenthesis x minus 1 right parenthesis left parenthesis x plus 4 right parenthesis end fraction
D. start fraction 14 over left parenthesis x minus 1 right parenthesis left parenthesis x plus 4 right parenthesis end fraction

Answer 17

[AI-generated answer — Bot]

B.

We need to find a common denominator for the two fractions. The first fraction already has a denominator of (x - 1), so we just need to multiply the second fraction by (x - 1)/(x - 1):

9/(x - 1) + 5/(x + 4) * (x - 1)/(x - 1) = 9/(x - 1) + 5(x - 1)/(x + 4)(x - 1)

Combining the two fractions, we get:

9/(x - 1) + 5(x - 1)/(x + 4)(x - 1) = [9(x + 4) + 5(x - 1)]/[(x - 1)(x + 4)]

Simplifying the numerator:

[9(x + 4) + 5(x - 1)] = (9x + 36 + 5x - 5) = 14x + 31

So the final simplified expression is:

[14x + 31]/[(x - 1)(x + 4)]

This is equivalent to answer choice B.

Answer 18

What is the solution to the equation The quotient 1 over x minus the quotient 6 over x squared equals negative 12?

A. x equals three fourths or x equals negative two thirds
B. x equals three fourths or x equals two thirds
C. x equals negative three fourths or x equals two thirds
D. x equals negative three fourths or x equals negative two thirds

Answer 19

[AI-generated answer — Bot]

B.

We can start by simplifying each quotient:

1/(x) - 6/(x²) = (x - 6x²)/[x³]

Multiplying both sides by x³, we get:

x - 6x² = -12x³

Bringing all terms to one side, we get a cubic equation in standard form:

12x³ - 6x² - x = 0

We can factor out x:

x(12x² - 6x - 1) = 0

Using the quadratic formula on the expression in the parentheses, we get:

x = [6 ± sqrt(6² + 4(12)(1))] / (2(12))

Simplifying:

x = [6 ± sqrt(60)] / 24 = [6 ± 2sqrt(15)] / 24

Reducing by a common factor of 2:

x = [3 ± sqrt(15)] / 12

Therefore, the solutions are:

x = (3 + sqrt(15)) / 12 or x = (3 - sqrt(15)) / 12

These solutions can be written as:

x = (1 + sqrt(15)) / 4 or x = (1 - sqrt(15)) / 4

This is equivalent to answer choice B.

Answer 20

Do the data in the table represent a direct variation or an inverse variation? Write an equation to model the data in the table.

x 1 3 5 10
y 4 12 20 40
A. direct variation; y equals 4 times x
B. direct variation; y equals one-fourth x
C. inverse variation; x times y equals 4
D. inverse variation; x times y equals one fourth

Answer 21

[AI-generated answer — Bot]

A.

We can determine whether the data represents a direct or inverse variation by checking if the ratio y/x is always the same. If it is, then the data represents a direct variation, and if it's the reciprocal of a constant, then it represents an inverse variation.

Checking the ratios:

y/x = 4/1 = 12/3 = 20/5 = 40/10 = 4

Since the ratio is always the same, we can see that it's a direct variation.

To write an equation for direct variation, we use the formula y = kx, where k is the constant of variation. We can determine the value of k by using one set of x and y values from the table. For example, using the first set of values, we get:

4 = k(1)

Solving for k:

k = 4

Therefore, the equation to model the data in the table is:

y = 4x

This is equivalent to answer choice A.

Answer 22

Which of the following points has coordinates that are 5 horizontal units from point A?

A coordinate grid is shown with point A located at left-parenthesis negative 2 comma 2 right-parenthesis.

Answer 23

[AI-generated answer — Bot]

The point that is 5 horizontal units from point A is:

(-7, 2)

We can find this point by moving 5 units to the left of point A along the x-axis. Therefore, the x-coordinate of the new point is -2 - 5 = -7, and the y-coordinate remains 2.

Answer 24

Find the distance between point A and point B.

A coordinate plane is shown with point A located at left-parenthesis negative 4 comma negative 3 right-parenthesis and point B located at left-parenthesis negative 4 comma 4 right-parenthesis.

Answer 25

[AI-generated answer — Bot]

The distance between point A and point B can be found using the distance formula:

distance = sqrt[(x2 - x1)² + (y2 - y1)²]

Plugging in the coordinates of points A (-4, -3) and B (-4, 4), we get:

distance = sqrt[(-4 - (-4))² + (4 - (-3))²] = sqrt[7²] = 7

Therefore, the distance between point A and point B is 7 units.

Answer 26

Which expression could be used to find the distance between point A and point B?

A coordinate plane is shown with point A located at left-parenthesis negative 4 comma 1 right-parenthesis and point B located at left-parenthesis 1 comma 1 right-parenthesis.

Answer 27

Which expression could be used to find the distance between point A and point B?

A coordinate plane is shown with point A located at left-parenthesis negative 4 comma 1 right-parenthesis and point B located at left-parenthesis 1 comma 1 right-parenthesis.